assuming "largest" is in terms of volume ...
sketch the circle $\displaystyle x^2 + y^2 = 2^2$
inscribe a rectangle in the circle ... rotation of the figure about the y-axis yields a cylinder inscribed in a sphere of radius 2.
volume of a cylinder, $\displaystyle V = \pi r^2 h$
$\displaystyle V = \pi x^2 \cdot 2y = \pi x^2 \cdot 2\sqrt{4 - x^2}$
find $\displaystyle \frac{dV}{dx}$ and find the values of x and y that maximize the volume ... then determine the surface area,
$\displaystyle A = 2\pi r^2 + 2\pi rh = 2\pi x^2 + 2\pi r (2y)$